Math 409/409GHistory of Mathematics

Pythagorean Triples

What’s a Pythagorean Triple?

A Pythagorean triple is a set of three positive integers x, y, and z which satisfy the Pythagorean theorem.

A Pythagorean triple is a set of three positive integers x, y, and z which satisfy the Pythagorean theorem.

y z

x

2 2 2x y z

We are all familiar with the well known 3-4-5 triangle, but have you ever tried to find less well known Pythagorean triples to use on a test and in a classroom demonstration?

If so, you most likely found such an endeavor frustratingly tedious, and will thus appreciate the what you are about to learn in this lesson.

Theorem

The integral solutions to are given by the formulas

where s and t are positive integers.

x y z2 2 2

2 2 2 22 , , and x st y s t z s t

Example

If you set s = 2 and t = 1, you get the 3-4-5 triangle.

With s = 3 and t = 2, you get the 12-5-13 triangle.

And so on.

2 2 2

2 2 2 2

for some positive integers and ,

2 , , and

x y z s t

x st y s t z s t

Proof

If

Then

2 2 2

2 2 2 2

for some positive integers and ,

2 , , and

x y z s t

x st y s t z s t

2 2 2 22 , , and x st y s t z s t 2 2 2 2 2 2

2 2 4 2 2 4

4 2 2 4

2 2 2

2

(2 ) ( )

4 2

2

( )

x y st s t

s t s s t t

s s t t

s t

z

On the other hand,

So and are reciprocals.

2 22 2 2 1

1

z yx y z

x x

z y z y

x x x x

z y

x x

z y

x x

Since x, y, and z are integers and since

and are reciprocals.

So there exist integers s and t such that

and

z y

x x

z y

x x

z y s

x x t

.z y t

x x s

and

Adding these equations results in

And subtracting them yields

z y s

x x t

.z y t

x x s

2 2 2 22

2

z s t s t z s t

x t s st x st

2 2 2 22

2

y s t s t y s t

x t s st x st

and

Since all the variables are integers, the numerators must be equal. The same goes for the denominators. This gives

as stated in the conclusion of the theorem.

2 2

2

z s t

x st

2 2

2

y s t

x st

2 2 2 22 , , and x st y s t z s t

So how does this help you?

At the beginning of this lesson I asked if you were ever frustrated in an attempt to find non-standard Pythagorean triples to use on tests or in classroom demonstrations.

Well this theorem tells you how to find them. In fact, I have written a program for the TI-83/84 that generates primitive Pythagorean Triples.

What’s a primitive Pythagorean triple?

A primitive Pythagorean triple is a Pythagorean triple where the greatest common divisor of all sides is 1.

For example: the 3-4-5 triangle is a primitive Pythagorean triple, but the 6-8-10 triangle is not primitive since all sides are divisible by 2.

Here’s what my TI-83/84 program displays.

Historical Background

The clay tablet “Plimpton 322” shows that the Babylonians used the formula in this theorem to generate Pythagorean triples somewhere between 1900 B.C. and 1600 B.C. That’s more than 1000 years before Pythagoras was born!

2 2 2

2 2 2 2

for some positive integers and ,

, , and

x y z s t

x st y s t z s t

Over 3200 years after the Babylonians, Pierre de Fermat (1601 – 1665) claimed that non-zero, integral solutions could be found for the general form of the Pythagorean theorem only when

Fermat stated that he had a truly wonderful proof of this, but it was too long to write in the margin!

2.n

n n nx y z

For over three centuries, mathematicians were unsuccessful in finding that “truly wonderful proof” Fermat claimed to have discovered. And no one could prove that he was wrong!

So Fermat’s claim that had non-trivial, integral solutions only when became known as Fermat’s Last Theorem.

n n nx y z

2n

Proving, or disproving, Fermat’s Last Theorem occupied the minds of numerous mathematicians for over 3 centuries.

It became known as one of the greatest unsolved problems in mathematics!

What’s the present status of Fermat’s Last Theorem?

Present Status of Fermat’s Last Theorem

In 1994, Andrew Wiles of Princeton University proved Fermat’s Last Theorem. His proof is approximately 100 pages long!

His proof deals with elliptic functions, an area of mathematics that was not in existence during Fermat’s time.

This ends the lesson on

Pythagorean Triples

Reference: CC Edwards, Pythagorean Triples, Eightysomething, January,

1999.